Cooperative Localization

Abstract—This paper describes an approach to cooperative
localization which finds its roots in robust estimation, employing
an unknown-but-bounded error model for sensor measurements.
In this framework, range and bearing measurements obtained
by the robots are viewed as constraints which implicitly
define a set of feasible solutions in the joint configuration space
of the robot team. The scheme produces bounded uncertainty
estimates for the relative configuration of the team by using
convex optimization techniques to approximate the projection
of this feasible set onto various subspaces of the configuration
space. The scheme can also be used to localize distributed sensor
nodes.

An important advantage of the proposed approach is that it is
able to produce bounded uncertainty estimates for the relative
configuration of the robots even in the case where the relative
orientations of the robots are completely unknown. This is an
important practical advance since errors in relative orientation
are often a major contributor to positioning uncertainty in
multi-robot localization schemes.

I. INTRODUCTION

Localization is a critical base level capability for mobile
robots and sensor ne tworks enabling numerous other
technologies including mapping, manipulation, and target
tracking. It is not surprising then that considerable research
effort has been directed at this problem [1], [2], [3], [4],
[5]. Within this realm of research , there is a narrower
yet still significant focus on cooperative localization for
multi-robot teams [6], [7], [8]. In this paradigm, groups of
robots combine sensor measurements to improve localization
performance. This approach is motivated by the fact that
robots within a team can often identify one another and
communicate sensor measurements, such as relative range
and bearing readings.

In this paper, we revisit our bounded uncertainty approach
to the multi-robot localization problem initially proposed
in [9]. Conceptually, the idea is that sensor measurements
induce constraints on the configuration space of the robot
team. Merging these constraints induces a feasible set on the
configuration space that represents the set of formation poses
that are consistent with all of the available sensor measurements.
Estimates for the uncertainty in various parameters of
C.J. Taylor is with the Department of Computer and Information Science,
University of Pennsylvania, 3330 Walnut St, Philadelphia PA. 19104, USA

J. Spletzer is with the Department of Computer Science and Engineering,
Lehigh University, 19 Memorial Drive West, Bethlehem PA. 18015,USA

This material is based upon work supported by the National Science
Foundation under Grant Nos. 9875867 and 0130858. Any opinions, findings,
and conclusions or recommendations expressed in this material are those of
the author(s) and do not necessarily reflect the views of the National Science
Foundation.
the team’s configuration such as the absolute position of a
single robot, or the relative positions of two or more nodes
can then be obtained by projecting this feasible set onto
appropriately chosen subspaces of the configuration space.
Unfortunately, recovering these projections exactly is quite
cumbersome - requiring exponential time in the number of
constraints. Instead, we propose a scheme to approximate
these projections using modern convex optimization techniques.
A significant shortcoming of our previous work was the
requirement that each of the robots have a sensor which
provided an orientation estimate with respect to a common
reference frame (e.g., a compass or solar sensor). Such an
as sumption makes the math more tractable , and is often
employed in multi-robot localization schemes [10], [11],
[12], [8]. The primary contribution of this work is a means
for localization using bearings sensors (e.g. cameras) that
can recover the relative robot positions without such an
orientation sensor. The approach can also accommodate
range measurements, and has a computational complexity
scaling polynomially in the number of robots. Furthermore,
the workload is readily distributed requiring only the communication
of sensor measurements between nodes.

The remainder of this paper is organized as follows:
Section II provides a literature review. A discussion of the
localization approach follows in Section III. Experimental
results are presented in Section IV. Finally, a discussion of
the approach’s merits along with directions for future work
are outlined in Section V.

II. RELATED WORK

In our previous work [9], we described bounded uncertainty
approaches that could be used in the case where
the robots could measure their orientation with respect to
a common frame of reference. In this case, the available
range and bearing measurements can be converted into linear
inequality constraints on the joint configuration space of the
robots. In this work we extend the same basic approach to
deal with feasible sets that are defined in terms of quadratic
constraints. This advance allows us to deal with upper
and lower bound range constraints and the relative bearing
constraints which will be described in the sequel. This in turn
allows us to handle cases where the relative orientations of
the robots are completely unknown.

This work relates to cooperative localization techniques,
where sensor measurements from multiple robots are integrated
to estimate uncertainties in absolute or relative position.
The first forays in this area employed direct localization
techniques, where the relative robot poses could be solved
analytically [13], [14]. These efforts were followed by other
researchers relying upon Bayesian techniques [6], [7], [12],
[8] for pose estimation. A primary advantage of Bayesian
approaches is that they not only propagate a state estimate
over time, but also the uncertainty associated with the state
estimate.

Our proposed approach also provides an estimate of the
uncertainty in the robot position. However, this is represented
instead in terms of feasible sets in the configuration space
and measurements are combined via set intersection. An
important advantage of this approach is that we do not
need to account for dependencies or correlations between
measurements to combine them using set intersection. We
can contrast this with the bookkeeping required to properly
account for dependencies between estimates in a Bayesian
framework. This is a particularly important advantage in the
context of cooperative localization where the measurements
of relative range and bearing induce a complex web of
dependencies among the node position estimates. Another
advantage of the approach is that it avoids the linearizations
inherent in approaches based on the Kalman filter. These
linearizations can be particularly problematic when they are
used to model the effects of uncertainties in the robots’ orientations.
In the sequel we will describe a bounded uncertainty
estimation scheme that can produce accurate estimates for
the robots configuration without any prior estimates for the
robots relative orientation. However, these benefits are not
without cost as the computational complexity exceeds that
of Kalman Filter approaches in both theory and practice.

The proposed approach shares the same general philosophy
as the set based SLAM algorithm proposed by Di
Marco, Garulli, Giannitrapani, and Antonio Vicino [21]. In
our work the feasible set is represented implicitly in terms of
a set of inequality constraints which is a significant point of
departure. It also explicitly deals with situations where the
robot orientations are unknown through the use of higher
order constraint equations on the feasible set.
Cooperative robot localization is also strongly related to
the problem of localizing wireless sensor nodes. This has
received significant attention in the sensor networks literature
and a number of effective approaches have been proposed
[15], [16], [17], [18], [19]. Perhaps most related is the work
of Biswas, Aghajan and Ye [18] who have also used relative
bearing measurements to help estimate the configuration of
sensor nodes [20]. Our work extends these approaches by
showing how relative bearing constraints can be used to
produce useful guaranteed bounds on the relative positions
of the node. Importantly, this allows us to tackle localization
problems that were previously beyond the reach of bounded
uncertainty techniques.

III. THE LOCALIZATION APPROACH

The basic elements of the localization framework are
diagrammed in Figure 1. Here a set of robots or sensor
nodes equipped with range and/or bearing sensors are located
on the plane. The edges in the figure indicate available

Fig. 1. A set of robot or sensor nodes located in a 2D workspace. The edges
between the nodes indicate available sensor measurements, that is an edge
between nodes i and j indicates that robot i can measure the range and or
bearing to node j from its current position.

Fig. 2. This smart camera node, which is equipped with an accelerometer,
can measure the relative bearing to other sensor nodes or robots.

relative measurements, that is an edge between nodes i and
j indicates that robot i can measure the range and or bearing
to node j from its current position. Note that the edges in
this graph will , in general, be directed.

For example, the proposed scheme can be employed to
localize ensembles of smart camera sensor nodes such as the
one shown in Figure 2. These nodes are capable of measuring
the relative bearing to other sensor nodes in the vicinity using
image measurements and their orientation with respect to
gravity using an onboard accelerometer [19]. Our method
can then be used to gauge the relative positions of the nodes
in the horizontal plane based on this information.
Let denote the configuration space of our robot
team. Let
denote the current configuration of the ensemble. Note that x
is simply the concatenation of the coordinates of the n robots .
Note also that we are purposely vague about the frame of
reference to which these coordinates are to be measured.
Depending on what parameters we are trying to estimate,
various choices will be more or less convenient. Hence we
defer this decision until the parameter estimation phase.

A. Generating Sensor Constraints
In this framework, we assume that we can bound the
error in the range and bearing measurements. Each such
measurement is then viewed as a constraint on the possible
values of the configuration vector x. This section focuses on

Fig. 3. Robot i simultaneously observes two other robots, j and k. The
measurement for the angle subtended at i can be used to constrain the
configuration of the team.

the transformations required to realize this.
In the sequel we will let the vector denote the
displacement between nodes i and j. This vector can be
expressed as a linear function of the configuration vector x as
follows: x, where denotes
the sparse projection matrix that extracts the coordinates of
node m from the configuration vector x in .

Importantly, all of the constraints that we will describe
in the following subsections can be expressed as quadratic
inequalities of the configuration vector x, that is they can
all be written in the form for some
. A constraint of this form
is termed convex if the matrix A is positive semidefinite
or 0 since inequalities of this form define convex regions
in the configuration space. Note that the set of quadratic
inequalities subsumes the set of all linear inequalities.
1) Relative Bearing Constraints: In Figure 3 Robot i
simultaneously observes two other nodes, j and k. The
vectors v_ij and v_ik denote the displacements between nodes
i and j and i and k respectively. Multiplying v_ij by the
constant rotation matrix yields a second
vector with the same magnitude as vij but rotated by 90
degrees counter clockwise. If we take inner products between
the vector v_ik and the vectors v_ij and we can form
another vector which encodes the angular separation , ,
between v_ij and v_ik:

The available bearing measurements would constrain this
vector to the sector defined by the two angles and
These constraints are expressed in the following inequalities.

Using Equation 1, both of these constraint equations can
be rewritten as quadratic functions of the parameter vector
x.

Note that these constraints will, typically, not be convex
functions of x. Note also that these constraints do not
require any knowledge of the robots orientation with respect
to an absolute frame of reference. This is an important
advance since it allows us to directly exploit the kinds of
relative bearing measurements that can be derived easily and
accurately from imaging sensors without requiring a compass
or any other orientation estimation scheme.

Relative bearing measurements have also been employed
in a different manner for bounded uncertainty single robot localization
problems by Briechle and Hanebeck [22]. Biswas,
Aghajan and Ye also make use of relative bearing constraints
in their work but use a different mathematical formulation
based on the circle defined by the three points. In contrast
our formulation provides equations which directly bound the
relative configuration of the nodes.

Fig. 4. Range measurements with bounded error constrain the distance
between two nodes i and j.

2) Range Constraints: Figure 4 depicts the annulus of
feasible configurations induced by constraints on the range
between nodes i and j. The constraints induced by a bounded
error range measurement can be expressed quite simply
[15] in terms of the configuration vector x. Here we again
note that the vector displacement between nodes i and j,
v_ij can be expressed as a linear function of the global
configuration vector x. i.e. . If and
respectively denote the upper and lower bounds on the
range measurement, we can derive two quadratic constraints
as follows:

Note that the upper bound constraint shown in Equation 6 is
a convex function of x while the lower bound constraint in
Equation 7 is concave.

Fig. 5. A convex approximation for the feasible set of displacements (the
dark region) can be obtained by bounding the its extent along various search
directions. The intersection of the resulting half planes defines a convex
polytope (the light gray area) which, by construction, encloses the feasible
set.

B. Gauging Team Configurations
Once a set of constraints has been derived from the
available measurements, we can proceed to consider the
problem of gauging the positions of the robots with respect to
each other. Consider for example the problem of estimating
the relative position of node j with respect to node i
represented by the displacement vector v_ij . As discussed
earlier, this vector can be expressed as a linear function of
the configuration vector x as follows: .
Hence, we can view the problem of bounding this vector
as one of gauging the projection of the set of feasible
configurations onto the 2D subspace corresponding to the
displacement v_ij .

We can bound the uncertainty in this displacement vector
by choosing search directions parallel to the subspace of
interest and bounding the extent of the feasible region in
those directions. This amounts to finding bounds for objective
functions of the form shown in Equation 8 given the
constraints derived from the measurements.

In this case the vector corresponds
to the search direction in the configuration space.

Figure 5 depicts the most general situation where the
feasible set and its projection need not be convex or even
connected. In any case, the goal of the localization scheme
is to produce a convex polyhedral approximation which
bounds the projection of all feasible configurations and,
hence, bounds the displacement vector. We see then that the
process of bounding the displacement vector relating two
nodes, vij , can be reformulated as a sequence of constrained
optimization problems which serve to gauge the extent of
the feasible region along various search directions. Here
we recognize that these optimization problems are actually
quadratically constrained quadratic programs (QCQP) since
both the objective function described in Equation 8 and the
constraint functions described in the previous sections can
be written as quadratic functions of the parameter vector x.
More precisely, our goal is to solve constrained optimization
problems of the following form.

Note that although the objective function described in equation
8 is actually a linear function of x, we are in fact able
to bound arbitrary quadratic functions of x. This means, for
example that we can use precisely the same machinery to
bound the square of the distance between nodes i and j
which can, clearly, be expressed as a quadratic function of
x as shown in Equation 10.

Most of the literature on solving QCQPs centers on the
special case where the constraint functions are convex which
is not, in general, the case here. Nonetheless we can make
progress by observing that there are convex relaxations of the
original optimization problem that provide useful bounds on
the feasible solutions. In this work we use the Lagrangian
relaxations described in [23], [24], [25]. The Lagrangian,
L(x,), and the Lagrangian dual, g(), of our original
optimization problem can be expressed as follows:

We can bound the minima of our original optimization
problem by finding the maximum of the Lagrangian dual
function g() over all non-negative values of . Note that
the Lagrangian dual will be convex even if the original
optimization problem is not. Using Shur complements this
optimization problem can be recast as a semidefinite program
as follows: maximize subject to the constraint that the
following matrix remains positive semi-definite.

We further constrain to be non-negative, that is
0, i = 1, . . . ,m. Once the problem is in this form, we can
apply modern semidefinite programming codes to solve this
optimization problem and provide a bound on the optimal
value. Importantly, these codes are able to exploit the sparse
structure of the A_i matrices to significantly reduce the
computational effort required to solve these problems.

1) Choosing Search Directions: Given the ability to
bound the extent of the feasible region in a given search
direction, we can now consider the question of how those
search directions should be chosen. A straightforward but
effective technique that works quite well in the general case
is to simply choose a set of uniformly spaced search directions
parallel to the plane of projection. With this approach,
the accuracy of the resulting convex approximation is simply
a function of the number of search directions employed.

In the special case where all of the measurement constraints
are convex we have proposed [9] a more sophisticated
approach to choosing search directions which allows us
to make precise statements about the performance of the
projection approximation scheme.

IV. LOCALIZATION EXPERIMENTS

The simulation experiments described in this section were
designed to characterize the performance of the localization
scheme under a variety of conditions. For each trial, a set of
10 nodes were randomly distributed on a plane. The x and
y coordinates of these locations were restricted to the unit
interval. A randomly chosen subset of the available range and
bearing measurements relating the nodes were considered.
These measurements were then corrupted with uniformly
distributed, bounded random errors. The range measurements
were corrupted with random errors in the range ±0.1 while
the bearing measurements were corrupted with errors in the
range ±0.25^◦.

The magnitude of the angular errors was chosen by
considering a typical camera with a 60 degree field of view
and a horizontal resolution of 640 pixels. In this context a
localization error of ±2 pixels on each bearing measurements
would translate to an error of approximately 0.2 degrees.
Note that this means that the relative bearing measurements
would have errors in the range ±0.4 degrees. Importantly,
we do not assume that the robots have access to an absolute
bearing sensor like a compass, so the robots do not, initially,
have any idea about their relative orientation.

The localization procedure described previously was used
to construct convex approximations for the position of each
of the nodes with respect to the first node which was fixed as
the origin (0, 0). To fix orientation and scale, the second point
in the set was also held fixed. The convex approximations
for the remaining 8 node locations were constructed by
considering 8 evenly spaced search directions in the plane.

For purposes of comparison, the localization procedure
was run under four different conditions. The first variant only
made use of the convex range constraints given in Equation
6 this experimental condition is analogous to the localization
scheme described in [15]. The second condition made use of
both the convex and non convex range constraints. The third
condition only considered the relative bearing constraints
while the last condition considered all available range and
bearing constraints.

The results obtained on a typical trial are shown in Figure
7. Figure 6 shows the ground truth configuration used for
this trial. Here the crosses denote the randomly chosen node
positions and the edges indicate the available range and
bearing measurements. A grand total of 51 range and bearing
measurements were used in this instance.

In order to compare the results of these four conditions
quantitatively, we computed the areas of the polyhedral

Fig. 6. For each trial a set of 10 node locations was chosen at random,
these are denoted by the blue crosses. The edges between the nodes indicate
available range and/or bearing measurements.
approximations returned by the procedure. These areas provide
an indication of how effective the constraints are at
narrowing down the set of feasible configurations. Larger
areas correspond to greater uncertainty. The results of this
analysis are summarized in Table I.

These results shows that under these experimental conditions,
the relative bearing constraints are much more powerful
than the range constraints. The convex approximations
constructed with these constraints are two orders of
magnitude smaller than those constructed using the range
constraints. When the range constraints are added to the
bearing constraints the average area is approximately halved.

In these experiments the results obtained with the range
only localization scheme reflect the fundamental flip ambiguity
associated with such measurements. That is, any
configuration that satisfies the range constraints can be
reflected in the plane to obtain another configuration that
also satisfies the constraints. The convex bounding region
that this method constructs must reflect that. In order to
overcome this, one would need to presuppose the existence
of known anchor points or some other means of resolving
the ambiguity. In this sense range and bearing measurements
can serve as complementary sources of information since
the bearing measurements can resolve the flip ambiguity
associated with the range measurements while the range
measurements resolve the scale ambiguity associated with
the bearing measurements.

In this round of simulation experiments the system did not
attempt to estimate the relative orientations of the robots. It
is possible to construct bounded uncertainty estimates for
these quantities when bearing measurements are available as
described in [26].

V. DISCUSSION AND CONCLUSIONS

This paper describes a novel approach to multi-robot
localization grounded in robust estimation. The scheme
employs an unknown-but-bounded error model for sensor
measurements and leverages recent advances in convex optimization
theory - specifically computational improvements

Fig. 7. The localization procedure was run under four conditions. a) Using only the convex range constraints b) Using both convex and non-convex range
constraints. c) Using only the relative bearing constraints d) Using all available range and bearing constraints. The blue crosses in these figures denote the
actual node locations while the red xs denote the centroids of the bounding polygons.
 

TABLE I
COMPARISON OF BOUNDING AREAS P⁺ OBTAINED UNDER VARIOUS EXPERIMENTAL CONDITIONS

in semidefinite programming techniques, duality, and Lagrangian
relaxations - to provide a localization framework
suitable for robotic systems.

Through convex approximations, our framework is able to
integrate any mixture of range and/or bearing measurements
from the robot formation into a single estimator which
provides estimates for the uncertainty in node positions
that are simultaneously conditioned on all available sensor
measurements. Since the method combines measurements
through set intersection rather than Bayes rules, it avoids
many of the issues associated with accounting for interdependencies
between multiple estimates. Furthermore, the
resulting uncertainty regions are guaranteed to contain the
true robot positions. Point estimates for the robots relative
positions can be obtained by considering the centroids or
Euclidean centers of various projections of the feasible set.

The ability to handle both convex and non-convex measurement
constraints is an important advance. With this
capability, we are able to make use of the relative bearing
constraint which takes the form of a nonconvex quadratic
inequality. This constraint is particularly useful since we
can typically measure relative bearings quite accurately with
imaging devices, often to within a small fraction of a degree .
Absolute bearing measurements, such as those obtained from
a compass are often off by a few degrees. Our localization
scheme is able to generate accurate estimates for the
configuration of a team of robots based on relative range
and bearing constraints without any prior estimates for the
robots orientation. This is potentially a significant advance
over EKF schemes which must account for uncertainties in
robot orientation through linearized approximations.

These advantages come at a cost however, and this is
the computational complexity of the associated semidefinite
programs. In theory, SDPs have an iteration complexity of
, where n corresponds to the number of nodes
and m the number of constraints - or in our case sensor
measurements [23]. However, it is well known that this
bound is conservative and based upon empirical results in
[15], we would expect the results to be closer to O(n³)
in practice. This makes the approach suitable for real-time
operations for reasonably large formation sizes (i.e., 10s of
robots).

While the computational complexity of our approach significantly
lags that of the EKF, it fares better with respect
to storage complexity. The EKF requires O(n²) storage
for representing the state covariance matrix. Assuming k
linear inqualities are used to model each uncertainty region,
we require O(kn) storage. However, our previous work in
[9] showed that the quality of the approximation of the
projection was independent of the number of nodes or sensor
measurements. As a result, O(n) storage complexity should
be expected.

While the proposed technique has been described in
the context of static nodes the method can be extended
to dynamic nodes by convolving the uncertainty regions
obtained at one instant with the uncertainties associated
with the motion model to derive additional constraints on
the node locations during subsequent timestep. We are also
investigating the problem of outlier rejection since incorrect
measurements with artificially small error bounds can result
in inconsistent constraints and an empty feasible set. Here
we may be able to employ techniques from robust estimation
like validation gates which test a proposed measurement
against an existing feasible set before acceptance. Another
approach would be to choose random subsets of the available
measurements to define the feasible set and identify outliers
[27].

VI. ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of an ARO
MURI DAAD-19-02-1-0383 “Adaptive Coordinated Control
of Intelligent Multi-Agent Teams (ACCLIMATE)”.

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